Poincaré conjecture

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The Poincare conjecture is a major theorem in algebraic topology first proposed in 1904 by Henri Poincaré and not proved until the solution was posted by Grigori Perelman on the internet beginning in 2002.

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Layman Statement of the Conjecture

An ordinary sphere (such as the surface of orange) is "simply connected" because a stretchable loop on it can be reduced to a single point without tearing it. A doughnut is an example of an object that is not simply connected.

The Poincare conjecture is this: the set of points at an equal radius from the center in four-dimensional space (known as a "three-dimensional sphere") is the only finite, bounded, simply connected object that locally looks like three-dimensional space and has the property that all loops and spheres within it can be shrunk to points.

Put another way, the conjecture states that if strings, surfaces, and other geometric objects "loop around" a three dimensional space in the same way they loop around what mathematicians call a "three sphere" (see below), then the space in question is a three sphere itself.

The Three Sphere

Mathematicians call the ordinary circle the "one sphere," which is written \mathbb{S}^1 \ . The defining quality of a circle is that it is the set of all points in plane which are the same distant from a specific point. Any deformation of this circle is a \mathbb{S}^1 \ - an object does not need to be a perfect circle to be called a one sphere by topologists, any closed loop which does not intersect itself is an example of \mathbb{S}^1 \ .

The surface of an ordinary sphere, like the Earth, is called a two sphere, or \mathbb{S}^2 \ . The defining quality of a sphere is that it is the set of all points in a Euclidean (ie, normal) three dimensional space which are the same distance from a set point. Note that the interior of a sphere is not \mathbb{S}^2 \ , and that like \mathbb{S}^1 \ , a deflated beach ball stretched into some strange shape is still \mathbb{S}^2 \ , so long as it does not self-intersect.

The three sphere, or \mathbb{S}^3 \ , refers to the set of all points in a four dimensional Euclidean space which are the same distance from a single point. Any deformation of that object is a \mathbb{S}^3 \ .

Strings, Surfaces, and Looping Around

For more information please see: Fundamental group and Homotopy group

By strings and surfaces, we mean the possible ways of embedding \mathbb{S}^1 \ and \mathbb{S}^2 \ into other spaces.

For the embedding of strings (\mathbb{S}^1 \ ) into two-dimensional spaces, it is easiest to imagine holding strings on these surfaces. For example, it is clear that any string we tie around a sphere, a basketball, for example, can be contracted until it is very small, even a point. While we can drape a string on a doughnut so that it can be shrunk to a point, there are ways we can loop strings around a doughnut so that they cannot - through the hole in the center, for example. Thus, we can distinguish between a sphere and a doughnut by noting that we can tie strings around them in different ways.

The natural question arises, if we CANNOT tie strings around two objects in different ways, are they the same object? This is, in essence, the Poincare conjecture, with the understanding that in addition to loops of string, we also discuss "tying" spheres around a space.

Formal Statement of the Conjecture

For those readers who have taken undergraduate topology, the statement of the Poincare conjecture is much shorter:

Let X \ be a 3 \ -dimensional manifold. Suppose that for every m \leq 3 \ , \pi_m(X) \cong \pi_m(\mathbb{S}^3) \ . Does it follow that X \ is homeomorphic to \mathbb{S}^3 \ ?

There is a more general form, which replaces 3 \ with n \ :

Let X \ be an n \ -dimensional manifold. Suppose that for every m \leq n \ , \pi_m(X) \cong \pi_m(\mathbb{S}^n) \ . Does it follow that X \ is homeomorphic to \mathbb{S}^n \ ?

Homeomorphism v. Diffeomorphism

For those readers who have some experience with topology, there may be a question of whether a diffeomorphism always exists. While the existence of a homeomorphism between manifolds demonstrates the existence of a diffeomorphism between them if they are of dimension \leq 3 \ , and it turns out that a diffeomorphism exists for n=5 \ and n=6 \ , but for most n\geq 7 \ it is possible to construct objects which are homeomorphic and not diffeomorphic - such objects are called "exotic \mathbb{S}^n." The only open case is the four dimensional one.

Proof of the Conjecture

The original statement of the conjecture went unproved until Gregori Perelman, a relatively unknown Russian mathematician, completed the proof which had eluded a century of expert mathematicians by demonstrating that no "cigar singularities" form in the Ricci flow on manifolds satisfying the conditions of the conjecture.

Easy Cases

While the original statement of the conjecture, also called the n=3 \ case, went unproved for a century, other versions were easily proved.

n=2 \ was proved even before the problem was stated by the easy classification of two-manifolds. Any senior-year college student who is majoring in mathematics should be able to prove the n=2 \ case.

n \geq 5 \ was proved by Steve Smale in 1961 with his h-cobordism theorem, for which he won a Fields Medal.

n=4 \ was proved in 1981 by Michael Freedman with his theorem on Casson handles, for which he shared a Fields medal with Andrew Casson, who had pioneered study in this area. Freedman's theorem was essentially a strengthening of the results of Smale's cobordism theorem, obtained at the expense of weakening the premises.

Unfortunately, the so-called "Whitney trick" which allowed the cobordism theorems of the late 20th century to produce such breakthroughs in topology is not possible in three dimensions, leaving Poincare's original formulation of the conjecture open.

Perelman's Proof

The proof of the conjecture relies on a concept called "Ricci flow," which is time-dependent deformation of a space which expands areas which are curved "negatively" (ie, two particles traveling in the same direction diverge) and contracts area which are curved "positively" (ie, two particles traveling in the same direction converge).

Two very important insights were necessary to turn the idea of Ricci flow into a proof of the Poincare conjecture. First, and this is the lesser of the insights, it was realized that no matter how complicated a manifold might be, the simple knowledge that a region of a three manifold could be bounded by a loop (or sphere) that could be shrunk to a point allows us to know for CERTAIN that Ricci flow on that region will either make it look like a patch of a three sphere after some finite "run time" for the flow, or "pinch" it beyond recognition (a singularity). This insight is due to Hamilton.

The second of these insights, which is the result of Perelman, is whenever the flow pinched the manifold, one could perform surgery on the manifold which would remove the pinch but leave the structure of the manifold unchanged otherwise.

The effect of the Ricci flow has much in common mathematically with the diffusion of heat in a surface, a concept called entropy. For example, look at the heat diffusion equation u_t = \Delta u \ . This equation is precisely an expression for the change in the metric of a manifold over time in Ricci flow if we take u = \ln g_{jj} \ , where j is any number less than the dimension of the manifold and no summation is implied on that index, and g is a metric symmetric on the main diagonal (any relevant manifold can have a coordinate system which create such a metric).

Noticing this (and other similarities between the effect of Ricci flow on a manifolds metric and the effect of time on heat dispersion, Perelman formulated the concept of "Perelman entropy."

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